AQA Paper 3 Specimen — Question 12 10 marks

Exam BoardAQA
ModulePaper 3 (Paper 3)
SessionSpecimen
Marks10
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Mark schemeDownload PDF ↗
TopicHypothesis test of binomial distributions
TypeOne-tailed hypothesis test (lower tail, H₁: p < p₀)
DifficultyStandard +0.8 This is a hypothesis testing question requiring setup of a binomial test with p=0.65, n=7, calculating P(X≤2) and comparing to 5% significance level. Part (b) requires understanding of binomial assumptions in context. While the mechanics are A-level standard, the one-tailed test setup, critical region determination, and contextual evaluation of assumptions require solid statistical reasoning beyond routine application.
Spec2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail
During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health. For the next 7 years, he only fell ill during 2 Christmas holidays.
  1. Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise. [6 marks]
  2. State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct. [4 marks]
Question 12:
AnswerMarks Guidance
12(a)States both hypotheses correctly
for one-tailed testAO2.5 B1
holidays without illness since
January 2007
( )
X ~ B 7,p
H p =0.65
0
H p<0.65
1
Under null hypothesis,
X B(7,0.65)
P(X ≤2)=0.0556
0.0556 >0.05
Accept H
0
There is not sufficient evidence
that the John’s rate of illness has
decreased
States model used (condone
AnswerMarks Guidance
0.009 rather than 0.056) PIAO1.1b M1
Using calculator, 0.056 or betterAO1.1b A1
Evaluates binomial model by
AnswerMarks Guidance
comparing P(X ≤2) with 0.05 PIAO3.5a M1
Infers H accepted PI
AnswerMarks Guidance
0AO2.2b A1
Concludes correctly in context.
‘not sufficient evidence’ or
AnswerMarks Guidance
equivalent requiredAO3.2a E1
(b)States one correct assumption(s)
regarding validity of modelAO3.5b E1
The probability of illness remains
constant throughout one’s life
Validity
Not fully valid, as age has an
impact on the immune system
OR
Assumption 2
Annual results (of illness) are
independent of one another
Validity
(Largely) valid. Trials are
sufficiently far apart that an
illness spanning two Christmases
is unlikely.
OR
Assumption 3
There are only two states, well
and ill
Validity
Unclear. Grey area exists.
eg does a mild sore throat count
as ill?
States corresponding correct
description(s) of likelihood of
AnswerMarks Guidance
validity in contextAO2.4 E1
States second correct
assumption(s) regarding validity of
AnswerMarks Guidance
modelAO3.5b E1
States corresponding correct
description(s) of likelihood of
AnswerMarks Guidance
validity in contextAO2.4 E1
Max two assumptions with
description of validity
AnswerMarks Guidance
Total10
QMarking Instructions AO 
Question 12:
--- 12(a) ---
12(a) | States both hypotheses correctly
for one-tailed test | AO2.5 | B1 | X = number of Christmas
holidays without illness since
January 2007
( )
X ~ B 7,p
H p =0.65
0
H p<0.65
1
Under null hypothesis,
X B(7,0.65)
P(X ≤2)=0.0556
0.0556 >0.05
Accept H
0
There is not sufficient evidence
that the John’s rate of illness has
decreased
States model used (condone
0.009 rather than 0.056) PI | AO1.1b | M1
Using calculator, 0.056 or better | AO1.1b | A1
Evaluates binomial model by
comparing P(X ≤2) with 0.05 PI | AO3.5a | M1
Infers H accepted PI
0 | AO2.2b | A1
Concludes correctly in context.
‘not sufficient evidence’ or
equivalent required | AO3.2a | E1
(b) | States one correct assumption(s)
regarding validity of model | AO3.5b | E1 | Assumption 1
The probability of illness remains
constant throughout one’s life
Validity
Not fully valid, as age has an
impact on the immune system
OR
Assumption 2
Annual results (of illness) are
independent of one another
Validity
(Largely) valid. Trials are
sufficiently far apart that an
illness spanning two Christmases
is unlikely.
OR
Assumption 3
There are only two states, well
and ill
Validity
Unclear. Grey area exists.
eg does a mild sore throat count
as ill?
States corresponding correct
description(s) of likelihood of
validity in context | AO2.4 | E1
States second correct
assumption(s) regarding validity of
model | AO3.5b | E1
States corresponding correct
description(s) of likelihood of
validity in context | AO2.4 | E1
Max two assumptions with
description of validity
Total | 10
Q | Marking Instructions | AO | Marks | Typical Solution
During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching.

In January 2007, he increased his weekly exercise to try to improve his health.

For the next 7 years, he only fell ill during 2 Christmas holidays.

\begin{enumerate}[label=(\alph*)]
\item Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise. [6 marks]

\item State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid.

For each assumption, comment, in context, on whether it is likely to be correct. [4 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3  Q12 [10]}}
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